Heat Equation

The module heateq can be used to solve the heat equation in 1D.

Getting Started

This module solve the partial differential (PDE) equation describing, for instance, the heat diffusion fenomena. The equation is as follows:

$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}$

The boundary condition is $\frac{\partial u}{\partial t}(-L, t) = \frac{\partial u}{\partial t}(L, t) = 0$ , where $L$ is the size of the sample.

Prerequisites

Make sure you have installed numpy and scipy in your systems or environment. It is suggested to install via pip:

pip install numpy
pip install scipy

Usage

In example bellow, the defalt sample is initiated: 10 mm linear sample (with spacing 0.1mm - equivalent to 100 points), heated at 100°C (372 K) and smoothly cooled down until both edges at 20ºC (296 K). The choosen material is iron (D = 23 mm²/s).

import heateq as heq
import matplotlib.pyplot as plt

D = 23  # for iron
dx = 0.1
u0, t, xscale = heq.initiate(dx=dx)
k = heq.kappa(u0, dx)  # The FFT frequencies (wavenumbers)

u = heq.solve(dudt, u0, t, D, k)

plt.figure(figsize=(8, 8))
plt.contourf(xscale, t, u, 100)
plt.colorbar()
plt.show()

Here's the sample image:

Built With

Numpy
scipy

Authors

Vagner Bessa - Initial work - bessava or bessavagner

License

This project is licensed under the MIT License - see the LICENSE.md file for details

Acknowledgments

IFCE - Instituto Federal de Ciência, Tecnologia e Educação do Ceara - Brazil
Inspiration: Steve Brunton See his chanel

bessavagner / heateq